Electron collisions with CO molecule: potential energy curves of higher lying CO− resonant states

Resonance energies and widths of eight long-lived metastable electronic states of the CO− anion are obtained using the R-matrix method as a function of bond distance. High-level ab initio scattering calculations are performed for a large number (above 150) of fixed-nuclear geometries using the large cc-pV6Z Gaussian basis set and a close-coupling model involving 27 low-lying target states. Potential curves for narrow resonances, three 2Σ+, four 2Π and one 2Δ, in the 10–14 eV region are reported, along with the data on the low-lying 2Π shape resonance. These curves provide a starting point for performing nuclear dynamics and hence studies of dissociative attachment via these states.


Introduction
There has been a long history of studies of electron-collision resonances in the CO molecule (Schulz 1973, Mazeau et al 1975, Wallbank et al 1983, Olszewski et al 1998. At low energies the CO 2 Π shape resonance at 1.6 eV is well known and found to lead to complicated structures in the observed cross sections (Haddad and Milloy 1983, Allan 1989, Gibson et al 1996, Poparić et al 2006, Allan 2010) which result in significant enhancements in the vibrational excitation cross sections (Campbell et al 2011, Laporta et al 2012. At higher energies (10-14 eV) studies have pointed towards a number of narrow resonances associated with the excited (Rydberg-like) electronic states of the CO molecule in this region (Mazeau et al 1975, Wallbank et al 1983, Newman et al 1983, Polley and Bailey 1988, Middleton et al 1993, Olszewski et al 1998. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
However, there has been little consensus between these studies over the nature or, indeed, number of the resonances.
These higher-lying resonances have also been extensively probed in dissociative electron attachment (DEA) experiments. These experiments monitor the C − or O − anions produced as result of an electron collision (Rapp and Briglia 1965, Chantry 1968, Stamatovic and Schulz 1970, Cadex et al 1975, Hall et al 1977, Nag and Nandi 2015b, a process that only occurs via a resonance (Fabrikant et al 2017). The observed DEA cross sections show a broad peak in the 9 to 12 eV energy range. Sanche and Schulz (1971) identified a 2 Σ + resonance at 10.04 eV which is assumed to contribute to this DEA peak. Furthermore, shape resonances at 10.4 eV and 10.7 eV and Feshbach resonances at 11.3 eV and 12.2 eV were reported from experiments (Schulz 1973, Sanche and Schulz 1971, Mazeau et al 1972; these may also contribute to DEA. DEA studies show that O − ion production is generally favoured and the cross section for C − production is small. Recent experiments have used velocity time sliced imaging as a method of probing the nature of these resonances through the measurement of O − angular distributions (Nag and Nandi 2015b, Tian et al 2013, Wang et al 2015. However, these studies arrived at different conclusions from analysing their results. Tian et al (2013) and Wang et al (2015) claimed to observe complete backward scattering of O − fragments and through their analysis using an axial recoil approximation model, they proposed that DEA at 10.6 eV proceeds through a coherent superposition of 2 Π, 2 Δ, and 2 Φ resonant states of CO − . However, Nandi (2015a, 2015b), who used a similar experimental setup to that used by Tian et al, questioned the above claim since they observed forward-backward asymmetry in their measurements of O − ion angular distribution. Using the same axial recoil approximation model, Nag and Nandi found that they could satisfactorily fit their data in terms of the more standard interference between different partial waves as opposed to the coherent superposition of different resonant states. The newer and more precise measurements made by Gope et al (2016) obtained results in close agreement to that of Nag and Nandi, which helped to settle the above controversy in favour of interference between different partial waves. However, theoretical calculations should be able to confirm the nature of the resonance(s) involved in this process.
A number of theoretical studies have identified resonances in the higher energy region. These include R-matrix calculations by Salvini et al (1984), who found a 2 Σ + shape resonance at about 20 eV, and by Morgan and Tennyson (1993), who found a rather broad (width, Γ > 1 eV) resonance for each of the 2 Σ + , 2 Π and 2 Δ symmetries they considered. Weatherford and Huo (1990) found 4 resonances in the 10-20 eV region based on a two-state calculation. Finally, Pearson and Lefebvre-Brion (1976) used the stabilization method to study continuum states of CO − ; they identified a single, narrow 2 Σ + symmetry resonance at 10.2 eV which corresponds with that observed by Sanche and Schulz (1971). More recently we (Dora et al 2016, Dora andTennyson 2019) have attempted to address this problem by performing R-matrix calculations using extended basis sets to represent the CO target and 27 CO target states in the close-coupling (CC) expansion. Our first study (Dora et al 2016) used both cc-pVDZ and a cc-pVTZ Gaussian type orbital (GTO) target basis set and up to 50 state in the CC expansion. The cc-pVDZ calculations did not recover any resonances above 10 eV while the cc-pVTZ ones found a narrow 2 Σ + resonance associated with the b 3 Σ + excited ('parent') target state. We concluded that to make progress on this problem we needed to further expand the basis set used to represent the target. So, in a preliminary study for the present work (Dora and Tennyson 2019), we tested a number of target representations and scattering models for CO at its equilibrium geometry. A calculation based on the use of cc-pV6Z GTO basis for the target found several narrow resonances, three 2 Σ + , one 2 Π and one 2 Δ, in the 10 to 13 eV range. These calculations form the starting point for the present work.
In this paper we present resonance energies and associated widths for CO anion states below 15 eV. These curves are used to identify parent states for the resonances and provide the necessary input for future DEA calculations.

Theory and calculation
Before starting scattering calculations it is necessary to perform calculations on the CO molecule to obtain target wavefunctions and energies, and target orbitals which form part of the input for the scattering calculation. Large numbers of target calculations were performed using the Molpro electronic structure code (Werner et al 2012). Calculations were performed at the complete active space self-consistentfield (CASSCF) level generally using state-averaging (SA-CASSCF). These calculations performed well in the region of the CO equilibrium geometry and in the asymptotic (dissociative) region but gave many difficulties at intermediate internuclear separations. In general these problems were caused by crossings both between individual potential curves which, for instance, caused them to move into and out of the chosen set of curves, and by the exchange of orbitals between those included in the active space and those not. Attempts to mitigate these problems by, for example, rotating the orbitals between different spaces (frozen, valence, virtual) to stop sharp exchanges and starting each calculation from the neighbouring geometry were only partially successful. In the end these issues forced us to limit the range of geometries for which we attempted to compute resonance curves.
For the scattering calculations we use the R-matrix method as implemented in the UK molecular R-matrix codes (UKRmol) (Morgan et al 1998, Carr et al 2012. A newer and upgraded version of the codes, called as UKRmol+ , became available as the current calculations were nearing completion. UKRmol+ can be used to study larger molecules, molecules at longer bond lengths and, pertinent to the present study, molecules with very diffuse target wavefunctions (Meltzer et al 2020). Below, we give a brief discussion of the method; the details of the principles involved can be found in the review article by Tennyson (2010).
In the R-matrix method the space around electron + target system is separated by an imaginary sphere of certain radius a. The size of the sphere is taken large enough so that all the N-electron target states have zero amplitude at the boundary. Inside the sphere, the N + 1-electron scattering wavefunction ψ N+1 k is represented by a CC expansion using the target states here, u i j are the continuum orbitals representing the scattering electron and A is the anti-symmetrization operator. The χ N+1 i , called L 2 configurations, are obtained by putting all N + 1 electrons in target molecular orbitals (MOs). These L 2 configurations are used to represent the scattering system at short range. At long range, outside the sphere, the outgoing electron is treated as interacting only with the multipolar potential field of the target molecule. The variational parameters a i jk and b ik are obtained from the diagonalization of the scattering Hamiltonian and are used in evaluating the R-matrix at the sphere boundary. The propagation of the R-matrix to large distances and matching with asymptotic solutions yields the K-matrix.
The eigenvalues of the K-matrix are related to the eigenphase sum; the later quantities can be fitted to a Breit-Wigner function to obtain the resonance parameters. In this work we have used this method as well as visual inspection of eigenphase sums and cross sections to find and fit the resonances in CO molecule as a function of geometry. As will be seen, most resonances lie very close to their parent target state which complicates the task of extracting resonance parameters.
Both, Molpro and the UKRmol codes cannot use the natural C ∞v symmetry of the CO molecule. Therefore, these calculations were performed using the reduced C 2v point group. Since, there are clear correlations linking electronic states between these two point groups, we report target and resonant states using C ∞v symmetry.
In this work we employ the largest compact basis set cc-pV6Z that Molpro can currently support. The need of such a large basis set was shown to be essential for representing the higher energy resonances in our previous work (Dora and Tennyson 2019) on electron-CO collisions at equilibrium geometry. The aug-cc-pV6Z basis set, which includes extra diffuse functions and so is better capable of representing higher lying resonances, was not employed as these basis functions tend to extend outside the R-matrix sphere, particularly at long internuclear separations. We note that a known consequence of the target basis leaking outside the box is the appearance of spurious extra resonances . Use of such a diffuse basis sets therefore requires larger, possibly much larger, R-matrix sphere size which leads to problems both with linear dependence in the continuum basis and greatly increased computational time. Such calculations were deemed impractical at the present time.
Initially, the CASSCF potential energy curves (PECs) of the above 27 target states were computed for internuclear bond distance R in the range of 1.5 R 4 a 0 on a fine grid of 0.001 a 0 . However, due to the difficulties associated with representation of higher-lying excited states and the multitude of avoided crossings among them, there were convergence problems at several geometries beyond R = 2.388 a 0 . Therefore, we present the PECs of the above 27 states of CO molecule for R 2.388 a 0 . The scattering calculations are also limited for R 2.388 a 0 for the above reason.

Scattering calculation
The R-matrix calculations have been performed for a total of 167 geometries comprising of 137 points in the R range of 1.7 a 0 to 2.38 a 0 on a closely spaced grid of 0.005 a 0 and at some additional nearby geometries where the calculations failed to fit resonances at the above chosen grid points. For the lowest 2 Π resonance we tried calculations extended up to R = 3.0 a 0 , but the resonance positions obtained did not behave smoothly beyond R = 2.5 a 0 due to the above mentioned problems associated with excited target states. The required target MOs for scattering calculations are taken from the saved target SA-CASSCF calculations described above which was performed for a finer R grid of 0.001 a 0 . Our preliminary investigations with cc-pV6Z basis set using both a = 12 a 0 and larger sphere size of 15 a 0 for the equilibrium geometry of CO showed little difference in the results.
In particular the two calculations identified the same number of resonances with those computed using the larger sphere lying just slightly higher in energy, which we attributed to the less complete representation of the continuum in this calculation. This finding is in line with our previous studies on larger molecules (Dora et al 2009). For reasons of computation resources and completeness of the continuum basis we  used the medium sphere defined by a = 12 a 0 for all scattering calculations presented here. The appropriate continuum MOs for this sphere size are obtained by partial wave expansion up to 4 which are represented by Gaussian-type functions (Faure et al 2002). Higher partial waves are known to be important to converge cross sections at the energies studied, see Zawadzki et al (2020) for discussion of this. However, the resonances are essentially short-ranged in nature and in this region the high behaviour is generally modelled by allowing the electron to enter unoccupied target orbitals. We therefore do not expect truncation of the partial wave expansion at g waves to have a significant impact on the resonance parameters. We note that the newly-developed UKRmol+ code  allows calculations to be performed with larger partial wave expansions. However, such calculations become computationally very demanding even for a molecule like H 2 (Meltzer et al 2020), and are currently not feasible with the large target basis set required for the present study.
The scattering wavefunction in the inner region is represented by close-coupling expansion by using the above mentioned 27 target states. The R-matrix calculated at the boundary of the sphere were propagated to a distance of 100 a 0 in order to match to the asymptotic analytic functions. We note that this model has also been used as part of a joint experimental-theoretical study on differential cross sections for electron impact electronic excitation of CO (Zawadzki et al 2020). The agreement between theory and experiment shown by this study was very good for the energy region considered here, which helps to validate our model.

Target PECs
Table 1 compares our calculated adiabatic electronic excitation energy, usually denoted T e , with the available experimental values. In general the agreement is good. Given that the resonances discussed below can all be associated with parent target states, the difference between the observed and calculated positions of the target states gives a measure of the uncertainty in our calculated resonance positions. The figure 1 shows all the 27 target potential energy curves used in the close coupling expansion of scattering wavefunctions in the R-matrix calculations.

Resonances
Beside the well-known, low-lying and broad 2 Π shape resonance, our calculations find eight other resonances, three with 2 Σ + symmetry, one of 2 Δ symmetry and a further four 2 Π Figure 2. Calculated CO −2 Π symmetry resonance curves. Also shown are the CO target states, see figure 1 for specifications of these states. All energies are given relative to the minimum of the X 1 Σ + CO ground state. symmetry resonances. These resonances are all narrow and lie in the 10-13 eV region; not all the resonances were detected at all geometries and in particular the 2 2 Π, 3 2 Π and 5 2 Π resonances were only detected for R 2 a 0 and are therefore not found at the CO equilibrium geometry. Table 2 presents a summary of these resonances while figures 2 and 3 show their behaviour as a function of internuclear separation.
It is common to associate Feshbach resonances with a particular excited state of the target which is usually described as the parent state. Figures 2 and 3 show that the narrow resonances do indeed all behave as if they are weakly bound to one or more of the CO excited states. These resonances can therefore all be classified as Feshbach in nature. Table 2 identifies parent states for each of these resonances. It can be seen that for some of the resonances more than one parent states is given. The 2 Δ resonance is found to follow both the 4 3 Π and 3 1 Π states. These states are nearly degenerate with potential curves that run parallel to each other at the geometries of interest. The 2 Δ can couple to both states via a p-wave and it is therefore reasonable to assume that these two states are joint parents. Conversely, the lowest 1 2 Σ + resonance appears to swap parents as function of internuclear separation: at short R it follows the a 3 Σ + curve which undergoes an avoided crossing with the b 3 Σ + curve at about 1.96 a 0 with the resonance following the lower curve and then after this curve crosses the b 3 Σ + curve at about 2.0 a 0 , it follows the b 3 Σ + curve. This parent swapping behaviour has previously been found in H − 2 Feshbach resonances (Stibbe and Tennyson 1997b). Similarly the second 2 Σ + resonance apparently disappears in the region of the same curve crossing. While this may be caused by the resonance becoming so weakly bound that our calculations failed to identify, the disappearance of resonances in such circumstance has been noted before (Stibbe and Tennyson 1998).
One reason for identifying the resonance parent states is that it allows an accurate position for each resonance to be determined even though the target calculation necessarily only gives approximate target curves. This is because the binding energy of a given resonance to its parent state is usually both small and appropriately constant as a function of R. This means that the main error in the resonance position arises from the target calculation and, if improved target energies are known, it is possible to adjust the resonance curve accordingly (Stibbe and Tennyson 1997a).
Figures 4-6 gives our resonance widths for the eight CO − resonances we identified and the lowest 2 Π shape resonance. The widths appear more structured than the resonance positions. There are a number of reasons for this. Resonance widths are very sensitive to details of the physics and experience shows that fitting is very robust in determining resonance positions but widths are sensitive to precise details of the fit. In order to obtain the resonance positions and widths as accurately as possible we performed outer region calculations on very fine energy grids close to each resonance. The number of energy grid points, grid size and the starting energy input required for each of these was estimated by looking at eigenphase plots of the standard set of initial calculations performed on 0.1 a 0 R grid. Moreover, for this initial set of calculations the resonance parameters were determined by using both Breit-Wigner fits to the eigenphase sums (Tennyson and Noble 1984) and the time delays (Little et al 2017) Figure 3. Calculated CO −2 Σ + and 2 Δ resonance curves. Also shown the CO target states, see figure 1 for specifications of these states. All energies are given relative to the minimum of the X 1 Σ + CO ground state. method; both these methods generally gave results close to each other. Therefore, for the later calculations in finer R grid only the eigenphase sums fitting method was used. At some of geometries the eigenphase sum fitting code (RESON) found multiple resonances some of which looked to be spurious, in such cases we picked the ones by looking at the eigenphase sum and cross section plots as well as tracing the continuity of resonance positions as function of R. Figures 4-6 show pronounced structures in the resonance widths. Some of these occur naturally due to crossings of the target curves and associated changes in parantage; some of the structure is undoubtedly an artifact of our calculation. As  shown in figure 4, the width of the 2 Π shape resonance drop almost monotonically with R; it become 0.11 eV at R = 2.6 a 0 and goes to zero as the anion state becomes bound relative the X 1 Σ + ground state of CO at R ≈ 2.7 a 0 . The curve, however, shows two jumps in these widths. These jumps are probably numerical, for example the structure at R = 1.95 a 0 appears to be associated with slight discontinuities in the in the underlying target electronic structure calculations at this geometry, rather than being due to any underlying physics of the problem. However, the jumps are relatively small and well within the uncertainty with which our calculation predicts this width. Conversely, there is pronounced structure in the 1 2 Σ + resonance width at about R = 2 a 0 , see figure 6; this occurs where the parent a 3 Σ + and b 3 Σ + states cross. The other variations are probably due to issues in performing fits for a narrow resonance close to a target curve. The structure in this resonance at short bondlengths is probably due to another curve cross although the density of target curves in this region makes it difficult to be definitive about which curves are involved. Similarly the structure in the width of the 2 Δ resonance at R < 1.8 a 0 , also shown in figure 6, is almost certainly associated with changes in target curves in this region, but the density of curves is such that it is difficult to be precise about this. The widths of the two very narrow resonances, the second and third 2 Σ + states, appear considerably busier. While some of this structure is also likely to be associated with changes in the target curves, the act of fitting leads to small differences in the widths which are exaggerated by the fact that these widths are magnified on the plot.

Conclusion
While it seems likely that many (maybe nearly all) molecules support multiple Feshbach resonances in the energy region just below the ionisation threshold; however, these have only been systematically characterised theoretically in the electron-H 2 system (Stibbe and Tennyson 1998). Here we present the first successful ab initio characterisation of CO resonances in the 10 to 14 eV region. These resonances are well-known from experimental studies and dissociative electron attachment (DEA) experiments have provided detailed insights into them. We find a significant number of narrow resonances, three 2 Σ + , four 2 Π and one 2 Δ, in the 10-14 eV region, as well as, of course, the broad low-lying 2 Π shape resonance. We have characterised the complex resonance potential energy curves (i.e., the positions and widths) for all these anion states as a function of internuclear separation. There are too many of these curves to make a direct association between individual resonance curves and particular DEA features. Theoretically, the next step is to use these complex resonance potential energy curves as input for DEA calculations. Similar studies have already been successfully performed for the 10-14 eV Feshbach H − 2 resonances , Celiberto et al 2013. To facilitate such calculations we provide our numerical values for the potential energy curves of our 27 target states and complex potential energy curves (positions and widths) for the 9 resonances we identify as supplementary data to this article.